Question

The prime factorization of a number is 3^2x5^3x7. Which statement is true about the factors of the number? ? Twenty-one is a factor of the number because both 3 and 7 are prime factors. Twenty-one is not a factor of the number because 21 is not prime. Ninety is a factor of the number because 3^2=9 and 90 is divisible by 9. Ninety is not a factor of the number because 90 is not divisible by 7.

Answer 1

The 1st statement is true.

21 is a factor because both 3 and 7 are factors.

Answer 2

Answer:

Twenty-one is a factor of the number because both 3 and 7 are prime factors.

Step-by-step explanation:

Given number is,

$3^2\times 5^3\times 7$

$=3\times 3\times 5\times 5\times 5\times 7$

Where, 3, 5 and 7 are prime numbers ( only divisible by 1 and itself ),

⇒ Both 3 and 7 are prime factors of the given number,

⇒ 21 is a factor of the given number.

Thus, first option is correct.

⇒ Second option is incorrect.

Now, 5 is factor of the given number but 2 is not,

⇒ 10 is not a factor of the given number,

⇒ 90 is not a factor of the given number,

⇒ Third option is incorrect.

Suppose 90 is divisible by 7,

⇒ 90 = 7a

Where a is any whole number,

⇒ $7=\frac{90}{a}$

$3^2\times 5^3\times 7=3^2\times 5^3\times \frac{90}{a}$

Since, 90 could be a factor of this number, if a = 3 or 5 or their multiple,

For the other values of a, 90 can not be the factor,

Hence, there is no effect of divisibility of 90 by 7 on having 90 as a factor of the given number,

⇒ Fourth option is incorrect.